# Vanishing of some Ext groups of coherent sheaves

We call a coherent sheaf 'of pure support' if it has no subsheaves with support of smaller dimension.

Now, let $X$ be a smooth projective variety, $F$ and $G$ coherent sheaves of pure support on $X$.

Is it true that $Ext^i(F,G)=0$ if $2\le n ={\rm dim \ supp}G- {\rm dim \ supp}F$ ?

This is related to the similar question Some questions on vanishing of Ext sheaves, where we have inequality in the latter equation, but we don't have the 'pure support' hypothesis.

Thank you very much.

• That is not true. It is not even true for a regular local ring $(R,\mathfrak{m})$ with residue field $k$. Using the Koszul complex (or Ischebeck's theorem, etc.), $\text{Ext}^i_R(R/\mathfrak{m},R)$ is zero except for $i =\text{dim}(R)$, in which case it equals $R/\mathfrak{m}$. On the other hand, for every $i$, $\text{Ext}^i_R(R/\mathfrak{m},R/\mathfrak{m})$ is the dual $k$-vector space of the $i^{\text{th}}$ exterior power of $\mathfrak{m}/\mathfrak{m}^2$. Thus, also $\text{Ext}^{i+1}_R(R/\mathfrak{m},\mathfrak{m}) = \text{Ext}^i_R(R/\mathfrak{m},R/\mathfrak{m})$ is nonzero. Commented Mar 20, 2017 at 9:51

Ischebeck's Theorem. For finitely generated modules $F$ and $G$ over a Noetherian local ring $(R,\mathfrak{m})$, $\text{Ext}_R^i(F,G)$ is the zero module for all $i < \text{depth}(G)-\text{dim}(\text{supp}(F))$.
Thus, for instance, if $G$ is a Cohen-Macaulay module of $\text{depth}(G)=\text{dim}(\text{supp}(G)) = d$, then it is true that $\text{Ext}^i_R(F,G)$ is the zero module for $i< \text{dim}(\text{supp}(G))-\text{dim}(\text{supp}(F))$. However, this can easily fail if $G$ is a module with small depth. For instance, if $(R,\mathfrak{m})$ is a Noetherian local ring of depth $d\geq 2$, then for $i < d$, $\text{Ext}^i_R(R/\mathfrak{m},R)$ is the zero module. Moreover, if $R$ is Gorenstein regular, then $\text{Ext}^d_R(R/\mathfrak{m},R)\to \text{Ext}^d_R(R/\mathfrak{m},R/\mathfrak{m})$ is injective. Thus, using the long exact sequence of Ext modules associated to the short exact sequence, $$\Sigma: \ \ \ \ \ 0\to \mathfrak{m} \to R \to R/\mathfrak{m} \to 0,$$ it follows that the connecting map $$\delta_\Sigma^i:\text{Ext}^i_R(R/\mathfrak{m},R/\mathfrak{m}) \to \text{Ext}^{i+1}_R(R/\mathfrak{m},\mathfrak{m}),$$ is an isomorphism for $i\leq d-2$, resp. for $i\leq d-1$ if $R$ is Gorenstein regular. (Edit. In fact, even in the nonregular case, $\delta_\Sigma^{d-1}$ is injective, so that $\text{Ext}^d_R(R/\mathfrak{m},\mathfrak{m})$ is nonzero whenever $\text{Ext}^{d-1}_R(R/\mathfrak{m},R/\mathfrak{m})$ is nonzero.)
If $R$ is a regular local ring of dimension $d$, then each $R$-module $\text{Ext}^i_R(R/\mathfrak{m},R/\mathfrak{m})$ is a vector space over $R/\mathfrak{m}$ of dimension $\binom{d}{i}$. So, in this case, for every $i$ with $1\leq j \leq d$, $\text{Ext}_R^j(R/\mathfrak{m},\mathfrak{m})$ is nonzero. In particular, the depth of $\mathfrak{m}$ equals $1$.
Presumably the reason for the hypothesis that $2\leq i$ is the evident nonzero element of $\text{Ext}^1_R(R/\mathfrak{m},\mathfrak{m})$ coming from the non-split short exact sequence $\Sigma$. However, even in the "best" case of regular local rings, $\text{Ext}_R^j(R/\mathfrak{m},G)$ can easily be nonzero for $R$-modules $G$ that have support of large dimension yet have small depth.